Lindeberg-Feller CLT is Theorem 4.12 in Foundations of Modern Probability by Olav Kallenberg that says
Let $ (ξn,j;j=1,…,n)^∞_{n=1}$ be a triangular array of row-wise independent random variables with mean $ 0$ and $\sum_{j=1}^n \mathbb{E}[\xi_{nj}^2] \to 1$ as $n→∞$ .
Suppose that for any ϵ>0 we have $ ∑_j E[ξ^2_{nj} ;|ξ_{nj} |>ε]→0 $ for all $ ε>0 $ (celebrated Lindeberg condition) .
Then$ \sum_{j=1}^n \xi_{n,j} \overset{d}{\to} N(0,1)$ .
from the definitions of book if (Ω,A,P) be the probability space
$E[ξ ;a]=∫_a ξ dp $ where $a ∈ A$
but I don't understand how $|ξ_{ni} |>ε$ can be our events.
if $ \\ξ_{n,j} = \frac{X_j - \mu_j}{\sqrt{\sum_{i = 1}^n \sigma_i^2}} $ how we can show that condition holds ?
